Convolution with an Hölder-function is an Hölder-function?

Question

Let $s\in(0,1)$, $n>2s$, $\Omega\subset\mathbb{R}^n$ open and bounded, let $u\in C^{0,2s+\epsilon}(\mathbb{R}^n)$, where $\epsilon>0$ is small, such that: $u=0$, on $\mathbb{R}^n\setminus\Omega$, let: $$ v(x):=\int_{\mathbb{R}^n}|y|^{2s-n}u(x-y)\,dy,\qquad\forall x\in\mathbb{R}^n,$$ is true that: $v\in C^{0,2s+\epsilon}(\mathbb{R}^n)$. The only thing that i have obtain is that, for any $x_1,x_2\in\mathbb{R}^n$, by hypothesis: $$ |u(x_1)-u(x_2)|\leq C|x-y|^{2s+\epsilon},$$ than: $$|v(x_1)-v(x_2)|\leq|x_1-x_2|^{2s+\epsilon}\int_{\mathbb{R}^n}|y|^{2s-n}\,dy, $$ but the problem is that: $$ \int_{\mathbb{R}^n}|y|^{2s-n}=\infty.$$ I have no idea how to proceed, any help would be appreciated.